Study of CAR-20/19-T Cells in Patients With Relapsed Refractory B Cell

Inclusion Criteria 1. Diagnosis of B-cell non-Hodgkin lymphoma (NHL) or chronic lymphocytic leukemia (CLL) / small lymphocytic leukemia (SLL): Patients must be aged≥18 years with relapsed, refractory disease and no available curative options that meet clinical criteria to initiate treatment. 2. Patients with B-cell NHL or CLL/SLL must have either CD19 or CD20 positive disease on most recent biopsy performed (a repeat biopsy is not mandatory for this study except as noted below). A minimum of 5% CD19 or CD20 positivity by immunohistochemistry or flow cytometry on prior or repeat biopsy is required. 3. Absolute CD3+ T cell count ≥50/mm\^3. 4. MRI brain and Lumbar Puncture with cerebral spinal fluid (CSF) analysis by cytology and flow cytometry without evidence of central nervous system (CNS) involvement only in patients with history of CNS involvement. 5. Measurable disease must have been documented within 4 weeks of the time of consent defined as the following by disease specific subtype: 1. B-cell NHL: Active disease defined as nodal lesions greater than 20 mm in the long axis or extranodal lesions \>10 mm in long and short axis or bone marrow involvement that is biopsy proven. 2. CLL/SLL: Active disease by either bone marrow, peripheral flow cytometry, or CT and/or positron emission tomography (PET) imaging with nodal disease. 6. Patients should have failed at least two lines of a standard treatment and meet disease specific criteria detailed below: 1. CLL/SLL: measurable disease as defined above that has relapsed after at least one line of chemo-immunotherapy and progressed or intolerant to ibrutinib monotherapy. 2. CD19 or CD20 positive B cell NHL limited to the following histologies: Advanced Stage III or IV Follicular Lymphoma, Diffuse Large B cell Lymphoma and associated subtypes (e.g. aggressive B-cell lymphoma, T-cell/histocyte rich B-cell lymphoma, primary mediastinal B-cell lymphoma, Epstein Barr virus positive diffuse large B-cell lymphoma, transformed lymphoma such as transformed follicular or marginal zone lymphoma, and Richter's transformation) and Mantle cell lymphoma. Specific criteria include: * Patients must have active, measurable disease after two lines of cytotoxic chemotherapy of which one must be anthracycline containing. * Must have received Rituximab or another CD20 antibody and at minimum two chemotherapy regimens appropriate for their disease. * Either failed autologous transplant or ineligible to receive autologous transplant 7. Karnofsky performance score ≥70. See Appendix A for scales. 8. Normal Baseline Neurological Evaluation: Mini-Mental Status Exam Score 24-30. 9. Adequate hepatic function, defined as aspartate aminotransferase (AST) and alanine aminotransferase (ALT) \<5 x upper limit of normal (ULN); serum bilirubin and alkaline phosphatase \<5 x ULN, or considered not clinically significant as per the clinical PIs discretion (e.g. Gilbert's or indirect hyperbilirubinemia) or felt to be due to underlying disease. 10. Adequate renal function, defined as creatinine clearance\>60 ml/min 11. Able to provide written informed consent. 12. Agree to practice birth control during the study. 13. Adequate cardiac function as indicated by New York Heart Association (NYHA) classification I or II AND left ventricular ejection fraction of ≥35% (by echocardiogram or MUGA) and adequate pulmonary function as indicated by room air oxygen saturation of ≥92%. 14. Expected survival \>12 weeks. 15. Negative urine or serum pregnancy test in females of child bearing potential at study entry and again within 48 hours' prior lymphodepleting chemotherapy. 16. Patients with prior blinatumomab treatment require repeat biopsy post-blinatumomab treatment that demonstrates CD19 or CD20 positive disease. 17. Meet criteria for regarding fertility and contraception. 18. Central line access will be required for CAR-20/19-T cell infusion. Exclusion Criteria 1. Positive beta-human chorionic gonadotropin in females of child-bearing potential. 2. Patients with known systemic allergy to bovine or murine products. 3. Known prior positive serology for human anti-mouse antibody (HAMA). 4. Confirmed active human immunodeficiency virus (HIV), Hepatitis B or C infection. 5. History of significant autoimmune disease OR active, uncontrolled autoimmune phenomenon: such as systemic lupus erythematous, Wegner's glomerulonephritis, autoimmune hemolytic anemia, idiopathic thrombocytopenic purpura (AIHA, ITP) requiring steroid therapy defined as \>20 mg of prednisone or equivalent daily. 6. Presence of ≥grade 3 non-hematologic toxicities as per CTCAE version 5.0 from any previous treatment unless it is felt to be due to underlying disease. 7. Concurrent use of investigational therapeutic agents or enrollment on another therapeutic clinical trial at any institution. Minimum of ≥4 weeks required from administration of any other investigational agents on other clinical trials prior to enrollment on this CAR-T protocol. 8. Refusal to participate in the long-term follow-up protocol. 9. Patients with active CNS involvement by malignancy on MRI or by lumbar puncture. a. Patients with prior CNS disease that has been effectively treated will be eligible providing treatment was \>4 weeks before enrollment and a remission documented within 8 weeks of planned CAR-T cell infusion by MRI brain and CSF analysis. 10. Previous recipients of allogeneic hematopoietic stem cell transplantation (AHCT) are excluded if they are \<100 days' post-transplant, have evidence of active graft-versus-host-disease (GVHD) of any grade, or are currently on immunosuppression. 11. Previous CAR-T cell therapy directed at either CD19 or CD20 within 100 days of planned CAR-20/19-T cell infusion (does not include re-enrollment) a. Patients with prior CAR-T treatment against CD19 or CD20 must have repeat biopsy confirming a minimum of 5% CD19 or CD20 positivity by immunohistochemistry or flow cytometry. 12. Anti-CD20 antibody treatment within 4 weeks of cell infusion. 13. Anti-CD19 antibody treatment within 4 weeks of cell infusion. 14. Cytotoxic chemotherapy other than lymphodepletion within 14 days of CAR-T cell infusion. 15. Cytotoxic chemotherapy treatment within 14 days or steroid treatment (other than replacement dose steroids) within 7 days prior to apheresis collection for CAR-T cells. 16. Patients post solid organ transplant who develop high grade lymphomas or leukemias. 17. Concurrent active malignancy other than basal or squamous cell carcinomas of the skin. Special Criteria for regarding Fertility and Contraception Female subjects of reproductive potential (women who have reached menarche or women who have not been post-menopausal for at least 24 consecutive months, i.e., who have had menses within the preceding 24 months, or have not undergone a sterilization procedure \[hysterectomy or bilateral oophorectomy\]) must have a negative serum or urine pregnancy test performed as part of eligibility criteria and again within 48 hours of initiation of lymphodepleting chemotherapy. Due to the high-risk level of this study, while enrolled, all subjects must agree not to participate in a conception process (e.g., active attempt to become pregnant or to impregnate, sperm donation, in vitro fertilization). Additionally, if participating in sexual activity that could lead to pregnancy, the study subject must agree to use reliable and double barrier methods of contraception during the follow-up period of the protocol. Acceptable birth control includes a combination of two of the following methods: * Condoms\* (male or female) with or without a spermicidal agent. * Diaphragm or cervical cap with spermicide * Intrauterine device (IUD) * Hormonal-based contraception Subjects who are not of reproductive potential (women who are premenarche or have been post-menopausal for at least 24 consecutive months or have undergone hysterectomy tubal ligation, salpingectomy, and/or bilateral oophorectomy or men who have documented azoospermia) are eligible without requiring the use of contraception.